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Soil properties, classification, compaction, permeability and shear strength fundamentals.
┌─────────────────────────────────────────────────────────────────┐
│ SOIL PHASE DIAGRAM │
├─────────────────────────────────────────────────────────────────┤
│ │
│ Soil is a 3-phase system: Solids + Water + Air │
│ │
│ ┌────────────────────────────────────┐ │
│ │ AIR (Va) │ Vv = Va + Vw │
│ ├────────────────────────────────────┤ │
│ │ WATER (Vw) │ │
│ ├────────────────────────────────────┤ │
│ │ SOLIDS (Vs) │ │
│ └────────────────────────────────────┘ │
│ V = Vs + Vw + Va = Vs + Vv = 1 (for unit volume basis) │
│ W = Ws + Ww (weight of air ≈ 0) │
│ │
│ VOLUME RELATIONSHIPS: │
│ Void Ratio: e = Vv / Vs │
│ Porosity: n = Vv / V = e / (1 + e) │
│ Degree of Satur.: S = Vw / Vv × 100% │
│ Air Content: a = Va / V = n(1 − S) │
│ │
│ WEIGHT RELATIONSHIPS: │
│ Water Content: w = Ww / Ws × 100% │
│ Bulk Density: γ = W / V │
│ Dry Density: γd = Ws / V = γ / (1 + w) │
│ Saturated Density: γsat = (Gs + e)γw / (1 + e) │
│ Submerged Density: γ' = γsat − γw = (Gs − 1)γw / (1 + e) │
│ Specific Gravity: Gs = γs / γw │
│ │
│ KEY RELATIONS: │
│ γ = (Gs + Se)γw / (1 + e) [general bulk density] │
│ γd = Gsγw / (1 + e) [dry density] │
│ e = wGs / S [for S = 100%: e = wGs] │
│ n = e / (1 + e) [porosity from void ratio] │
│ e = n / (1 − n) [void ratio from porosity] │
│ │
│ TYPICAL VALUES: │
│ Gs: Sand = 2.65, Silt = 2.67, Clay = 2.70-2.80 │
│ γw = 9.81 kN/m³ ≈ 10 kN/m³ │
│ γ (bulk): Sand = 16-20, Clay = 16-22 kN/m³ │
│ γd (compacted): 15-22 kN/m³ │
└─────────────────────────────────────────────────────────────────┘| Property | Symbol | Test | Significance |
|---|---|---|---|
| Water Content | w (%) | Oven drying (105°C) | Strength & compressibility indicator |
| Specific Gravity | Gs | Pycnometer | Phase calculations |
| Grain Size Distribution | D10, D30, D60 | Sieve + Hydrometer | Classification (Cu, Cc) |
| Liquid Limit | LL | Casagrande device | Plasticity range upper bound |
| Plastic Limit | PL | Thread rolling (3mm) | Plasticity range lower bound |
| Shrinkage Limit | SL | Mercury/dry method | Volume change behavior |
| In-Situ Density | γ | Core cutter / Sand replacement | Compaction control |
| Parameter | Formula | Criteria |
|---|---|---|
| Cu (Uniformity Coeff.) | D60 / D10 | > 6 for well-graded gravel; > 4 for sand |
| Cc (Curvature Coeff.) | (D30)² / (D10 × D60) | 1 ≤ Cc ≤ 3 for well-graded |
e = wGs/S. Submerged unit weight γ' = γsat − γw is the critical parameter for effective stress calculations below the water table.| Limit | Definition | Transition | Test Method |
|---|---|---|---|
| Liquid Limit (LL) | Water content at which soil flows | Liquid ↔ Plastic | Casagrande device (25 blows) |
| Plastic Limit (PL) | Water content at 3mm thread crumbles | Plastic ↔ Semi-solid | Thread rolling method |
| Shrinkage Limit (SL) | Water content below which no volume decrease | Semi-solid ↔ Solid | Mercury method |
| LI Value | Consistency | State |
|---|---|---|
| LI < 0 | Stiff / Hard | w < PL; semi-solid or solid |
| 0 ≤ LI ≤ 1 | Plastic | PL ≤ w ≤ LL; normal range |
| LI > 1 | Liquid | w > LL; sensitive or quick clay |
| LI = 1 | At Liquid Limit | Boundary between plastic and liquid |
┌─────────────────────────────────────────────────────────────────┐
│ IS CLASSIFICATION (IS: 1498-1970) │
├─────────────────────────────────────────────────────────────────┤
│ │
│ COARSE-GRAINED (Sieve #475 µm > 50% retained): │
│ ┌────────────────────────────────────────────────────────┐ │
│ │ GRAVEL (G): > 50% retained on 4.75 mm sieve │ │
│ │ GW Well-graded gravel Cu>6, Cc=1-3 │ │
│ │ GP Poorly-graded gravel Cu<6 or Cc outside 1-3 │ │
│ │ GM Silty gravel Fines 5-12%, PI < 4 │ │
│ │ GC Clayey gravel Fines > 12%, PI > 7 │ │
│ │ │ │
│ │ SAND (S): > 50% passes 4.75 mm sieve │ │
│ │ SW Well-graded sand Cu>6, Cc=1-3 │ │
│ │ SP Poorly-graded sand Cu<6 or Cc outside 1-3 │ │
│ │ SM Silty sand Fines 5-12%, PI < 4 │ │
│ │ SC Clayey sand Fines > 12%, PI > 7 │ │
│ └────────────────────────────────────────────────────────┘ │
│ │
│ FINE-GRAINED (Sieve #475 µm > 50% passing): │
│ ┌────────────────────────────────────────────────────────┐ │
│ │ Use Casagrande's Plasticity Chart: │ │
│ │ A-line: PI = 0.73(LL − 20) │ │
│ │ Above A-line: CL, CH │ │
│ │ Below A-line: ML, MH │ │
│ │ LL < 35: Low plasticity (L) │ │
│ │ 35 ≤ LL ≤ 50: Intermediate (I) │ │
│ │ LL > 50: High plasticity (H) │ │
│ │ OL: Organic silt of low plasticity (LL < 50, below A) │ │
│ │ OH: Organic clay of high plasticity (LL > 50, below A) │ │
│ │ Pt: Peat / highly organic soil │ │
│ └────────────────────────────────────────────────────────┘ │
│ │
│ ──────────────────────────────────────────────────────────── │
│ USCS (Unified Soil Classification System): │
│ ──────────────────────────────────────────────────────────── │
│ Same as IS system but developed by Casagrande (USA). │
│ Key difference: USCS uses dual symbols (e.g., SC-SM) │
│ when fines are between 5% and 12%. │
│ │
│ HIGHLY ORGANIC SOILS (both systems): │
│ Pt → Peat; dark color, organic odor, fibrous texture │
│ Identified by: Loss on ignition, visual inspection │
└─────────────────────────────────────────────────────────────────┘PI > 0.73(LL − 20), it's clay. Organic soils (OL, OH) plot below the A-line AND have lower LL upon oven drying (confirm with oven test).┌─────────────────────────────────────────────────────────────────┐
│ SOIL COMPACTION — THEORY & PROCTOR TEST │
├─────────────────────────────────────────────────────────────────┤
│ │
│ COMPACTION: Process of densifying soil by applying mechanical │
│ energy (removes air, not water). │
│ │
│ PROCTOR COMPACTION TEST: │
│ ───────────────────────── │
│ Standard Proctor (IS: 2720 Part VII): │
│ Mold: 1000 cc (10 cm diameter, 12.7 cm height) │
│ Hammer: 2.6 kg, 310 mm drop │
│ Layers: 3 │
│ Blows/layer: 25 │
│ Energy: 600 kN-m/m³ │
│ │
│ Modified Proctor (IS: 2720 Part VIII): │
│ Mold: 1000 cc │
│ Hammer: 4.89 kg, 450 mm drop │
│ Layers: 5 │
│ Blows/layer: 25 │
│ Energy: 2670 kN-m/m³ (4.5× standard) │
│ │
│ COMPACTION CURVE: │
│ Plot: Dry density (γd) vs Water content (w%) │
│ Peak → MDD (Maximum Dry Density) + OMC (Optimum Moisture Content)│
│ │
│ Zero Air Void Line (ZAVL): │
│ γd = Gsγw / (1 + wGs) [S = 100%] │
│ Curve on right side; no soil can be to the right of ZAVL │
│ │
│ EFFECT OF COMPACTION ENERGY: │
│ Higher energy → Higher MDD, Lower OMC │
│ Curve shifts up and to the left │
│ │
│ RELATIONSHIPS: │
│ Air content at OMC: usually 3-5% │
│ Compacted γd > 95% of MDD (field specification) │
│ w_field = OMC ± 2% (acceptable range) │
│ │
│ FIELD COMPACTION CONTROL: │
│ 1. Sand replacement method (large pits) │
│ 2. Core cutter method (soft soils) │
│ 3. Rubber balloon method │
│ 4. Nuclear density gauge (fast, in-situ) │
│ 5. Dynamic cone penetrometer (compaction verification) │
└─────────────────────────────────────────────────────────────────┘| Factor | Effect | Best Practice |
|---|---|---|
| Moisture Content | Peak density at OMC | Compact at OMC ± 2% |
| Compaction Energy | ↑ Energy → ↑ MDD, ↓ OMC | Use appropriate effort for design |
| Soil Type | Clay needs more water than sand | Coarse soils: vibration; Fine: kneading |
| Gradation | Well-graded compacts better | Add fines to fill voids in gap-graded |
| Layer Thickness | Thinner = better compaction | Max 200-300 mm per layer |
| Number of Passes | ↑ Passes → ↑ density (diminishing) | Optimize passes (typically 4-8) |
| Roller Speed | Slower = better compaction | 2-6 km/h optimal |
| Equipment | Best For | Mechanism |
|---|---|---|
| Smooth Wheel Roller | Base courses, crushed rock | Static pressure |
| Pneumatic-Tired Roller | Base, sub-base, uniform压实 | Kneading + pressure |
| Sheepfoot Roller | Clay, cohesive soils | Kneading action (penetration) |
| Vibrating Roller | Granular soils, sand | Vibration + pressure |
| Grid Roller | Coarse-grained, weathered rock | Crushing + pressure |
| Tamping Roller | Cley fills, earth dams | Impact compaction |
┌─────────────────────────────────────────────────────────────────┐
│ PERMEABILITY — DARCY'S LAW & SEEPAGE │
├─────────────────────────────────────────────────────────────────┤
│ │
│ DARCY'S LAW: │
│ v = k × i (velocity of flow through soil) │
│ Q = k × i × A (discharge through area A) │
│ │
│ Where: │
│ v = discharge velocity (superficial) │
│ k = coefficient of permeability (hydraulic conductivity) │
│ i = hydraulic gradient = h/L │
│ h = head loss, L = length of flow path │
│ A = cross-sectional area of soil specimen │
│ │
│ SEEPAGE VELOCITY (actual pore velocity): │
│ v_s = v / n = Q / (A_v) │
│ where n = porosity, A_v = A × n (area of voids) │
│ v_s > v (actual velocity is higher than Darcy velocity) │
│ │
│ TYPICAL PERMEABILITY VALUES: │
│ ┌──────────────────────┬──────────────┐ │
│ │ Soil Type │ k (cm/sec) │ │
│ ├──────────────────────┼──────────────┤ │
│ │ Gravel │ 1.0 - 10.0 │ │
│ │ Coarse Sand │ 0.1 - 1.0 │ │
│ │ Fine Sand │ 10⁻²-10⁻³ │ │
│ │ Silt │ 10⁻³-10⁻⁶ │ │
│ │ Clay │ 10⁻⁶-10⁻⁹ │ │
│ └──────────────────────┴──────────────┘ │
│ │
│ LABORATORY TESTS: │
│ Constant Head Test: k = QL / (Aht) [for coarse soils] │
│ Falling Head Test: k = (aL/At) × ln(h1/h2) [for fine soils]│
│ │
│ FIELD TESTS: │
│ Pumping-out test (unconfined): │
│ k = Q / (π × r²w × sy) where r = radial distance │
│ sy = drawdown at distance r │
│ Pumping-out test (confined): │
│ k = Q / (2π × b × Δs) b = aquifer thickness │
│ │
│ PERMEABILITY RELATIONS: │
│ k ∝ D²₁₀ (Hazen's formula for clean sand) │
│ k = C × D²₁₀ × (T/10)°C [C ≈ 100, D₁₀ in cm] │
│ Kozeny-Carman: k = (γw/μ) × (e³/(1+e)) × (1/Ss²) × Ck │
└─────────────────────────────────────────────────────────────────┘┌─────────────────────────────────────────────────────────────────┐
│ FLOW NETS │
├─────────────────────────────────────────────────────────────────┤
│ │
│ Flow net = combination of flow lines (ψ) and equipotential │
│ lines (φ) that form an orthogonal grid under Laplace equation: │
│ │
│ ∂²h/∂x² + ∂²h/∂y² = 0 (Laplace equation for 2D flow) │
│ │
│ PROPERTIES OF FLOW NET: │
│ ┌──────────────────────────────────────────────────────────┐ │
│ │ 1. Flow lines and equipotential lines intersect at 90° │ │
│ │ 2. Flow channels between adjacent flow lines carry equal Q│ │
│ │ 3. Equipotential drops are equal between adjacent lines │ │
│ │ 4. Curvilinear squares: Δs ≈ Δn (ideally) │ │
│ └──────────────────────────────────────────────────────────┘ │
│ │
│ SEEPAGE CALCULATION: │
│ Q = k × h × (Nf / Nd) × (W/W) │
│ Q = k × H × (Nf / Nd) │
│ │
│ Where: │
│ Nf = number of flow channels │
│ Nd = number of equipotential drops │
│ H = total head loss │
│ k = permeability │
│ │
│ EXIT GRADIENT (critical for piping): │
│ i_exit = Δh / ΔL (gradient at exit face) │
│ i_critical = (Gs − 1) / (1 + e) = (γsat − γw) / γw │
│ Factor of Safety = i_critical / i_exit ≥ 4-5 │
│ │
│ QUICK SAND CONDITION: │
│ Occurs when upward seepage force balances effective weight │
│ i = i_critical → effective stress σ' = 0 → soil boils │
│ This is also the condition for BOILING / PIPING │
│ │
│ LICENSING FILTER CRITERIA (Terzaghi): │
│ D₁₅(filter) / D₈₅(base soil) ≤ 4-5 (piping criterion)│
│ D₁₅(filter) / D₁₅(base soil) ≥ 4-5 (permeability) │
│ D₅₀(filter) / D₅₀(base soil) ≤ 25 │
└─────────────────────────────────────────────────────────────────┘(Gs−1)/(1+e). For typical Gs = 2.65 and e = 0.65, i_critical ≈ 1.0. To prevent piping beneath dams, use FS = i_critical / i_exit ≥ 4-5.┌─────────────────────────────────────────────────────────────────┐
│ TERZAGHI'S 1D CONSOLIDATION THEORY │
├─────────────────────────────────────────────────────────────────┤
│ │
│ CONSOLIDATION: Time-dependent volume change of saturated clay │
│ due to expulsion of pore water under sustained load. │
│ │
│ STAGES OF COMPRESSION: │
│ 1. Initial: Δσ applied → u increases (undrained) │
│ 2. Transition: Δσ gradually transfers to σ' (drainage) │
│ 3. Final: All Δσ carried by σ' (fully drained) │
│ Δσ = Δσ' + Δu at all stages (Terzaghi's principle) │
│ │
│ CONSOLIDATION PARAMETERS: │
│ Cc = Compression Index (virgin curve slope in e-logσ' plot)│
│ Cr = Recompression Index (reload slope) │
│ Cs = Swelling Index (unloading slope) │
│ Pc = Pre-consolidation pressure (max past pressure) │
│ Cv = Coefficient of consolidation │
│ Cα = Coefficient of secondary compression │
│ av = Coefficient of compressibility = −Δe/Δσ' │
│ mv = Coefficient of volume compressibility = av/(1+e₀) │
│ │
│ SETTLEMENT CALCULATION: │
│ ───────────────────────── │
│ For NC clay (σ'₀ + Δσ ≤ Pc): │
│ Sc = Cc × H / (1+e₀) × log₁₀((σ'₀ + Δσ)/σ'₀) │
│ │
│ For OC clay (σ'₀ + Δσ ≤ Pc): │
│ Sc = Cr × H / (1+e₀) × log₁₀((σ'₀ + Δσ)/σ'₀) │
│ │
│ For OC clay (σ'₀ < Pc < σ'₀ + Δσ): │
│ Sc = Cr × H/(1+e₀) × log₁₀(Pc/σ'₀) │
│ + Cc × H/(1+e₀) × log₁₀((σ'₀+Δσ)/Pc) │
│ │
│ Where H = thickness of clay layer, e₀ = initial void ratio │
│ σ'₀ = initial effective stress, Δσ = increase │
│ │
│ TIME RATE OF CONSOLIDATION: │
│ Tv = Cv × t / d² │
│ d = drainage path = H/2 (double drainage) │
│ d = H (single drainage) │
│ │
│ For U ≤ 60%: Tv = (π/4) × U²/100 │
│ For U > 60%: use Taylor's square root or log methods │
│ │
│ At U = 90%: Tv = 0.848 │
│ t₉₀ = 0.848 × d² / Cv │
│ │
│ SECONDARY COMPRESSION: │
│ Ss = Cα × H × log₁₀(t₂/t₁) │
│ Cα ≈ 0.005-0.02 for inorganic clays │
│ Cα ≈ 0.02-0.05 for organic clays │
└─────────────────────────────────────────────────────────────────┘| Parameter | From e-log σ' Plot | Typical Range |
|---|---|---|
| Pc (Preconsolidation) | Intersection of initial & virgin curves | Variable (past loading history) |
| Cc (Compression Index) | Slope of virgin line | 0.1-0.5 (clays), 0.005-0.025 (sands) |
| Cr (Recompression Index) | Slope of reload line | 0.015-0.05 |
| Cv (m²/year) | Square root or log fitting | 10⁻⁴-10⁻¹ |
| OCR | Pc / σ'₀ | NC: ~1.0; OC: > 1.0; UC: < 1.0 |
| OCR Value | Classification | Behavior |
|---|---|---|
| OCR = 1 | Normally Consolidated (NC) | Settlement follows virgin curve |
| OCR > 1 | Over-Consolidated (OC) | Settlement follows recompression then virgin |
| OCR < 1 | Under-Consolidated (UC) | Still consolidating under past load |
Cc ≈ 0.009(LL − 10)(Skempton's correlation). Always check OCR first to determine whether the soil is NC or OC before choosing the correct settlement formula.┌─────────────────────────────────────────────────────────────────┐
│ MOHR-COULOMB FAILURE CRITERION │
├─────────────────────────────────────────────────────────────────┤
│ │
│ MOHR-COULOMB EQUATION: │
│ τf = c + σ' × tan(φ) │
│ │
│ Where: │
│ τf = shear strength at failure │
│ c = cohesion intercept (kPa) │
│ σ' = effective normal stress on failure plane │
│ φ = angle of internal friction (degrees) │
│ │
│ IN TERMS OF TOTAL STRESS (UU condition): │
│ τf = cu (φ = 0 condition, undrained clay) │
│ cu = undrained shear strength │
│ │
│ EFFECTIVE STRESS FORM: │
│ τf = c' + σ' × tan(φ') │
│ σ' = σ − u (Terzaghi's effective stress principle) │
│ │
│ STRESSES ON FAILURE PLANE: │
│ θf = 45° + φ/2 (angle of failure plane from major plane) │
│ σ₁ = σ₃ × tan²(45° + φ/2) + 2c × tan(45° + φ/2) │
│ σ₃ = σ₁ × tan²(45° − φ/2) − 2c × tan(45° − φ/2) │
│ │
│ TYPICAL VALUES: │
│ ┌──────────────────┬────────────┬───────────┐ │
│ │ Soil Type │ c' (kPa) │ φ' (degrees)│ │
│ ├──────────────────┼────────────┼───────────┤ │
│ │ Loose Sand │ 0 │ 28-32 │ │
│ │ Dense Sand │ 0 │ 35-45 │ │
│ │ Soft Clay (UU) │ 25-50 │ 0 (φ=0) │ │
│ │ Stiff Clay │ 50-100 │ 15-25 │ │
│ │ Saturated Clay │ c' only │ 0 (φ=0) │ │
│ └──────────────────┴────────────┴───────────┘ │
│ │
│ SHEAR STRENGTH TESTS: │
│ ┌──────────────────┬────────────┬──────────────────────────┐ │
│ │ Test Type │ Drainage │ Parameter Measured │ │
│ ├──────────────────┼────────────┼──────────────────────────┤ │
│ │ UU (Unconsolid- │ Unconsol- │ cu, φu = 0 │ │
│ │ ated Undrained) │ idated, │ (total stress analysis) │ │
│ │ │ Undrained │ │ │
│ │ CU (Consolidated │ Consolid- │ c', φ' (total + effective)│ │
│ │ Undrained) │ ated, │ via pore pressure meas. │ │
│ │ │ Undrained │ │ │
│ │ CD (Consolidated │ Consolid- │ c', φ' (effective stress) │ │
│ │ Drained) │ ated, │ long-term analysis │ │
│ │ │ Drained │ │ │
│ └──────────────────┴────────────┴──────────────────────────┘ │
│ │
│ DIRECT SHEAR TEST: │
│ τf = P/A; σ = N/A │
│ Normal stiffness: horizontal displacement controlled │
│ Fast = UU; Slow = CD │
└─────────────────────────────────────────────────────────────────┘| Parameter | Symbol | Determination |
|---|---|---|
| Cohesion | c or c' | Y-intercept of Mohr-Coulomb envelope |
| Friction Angle | φ or φ' | Slope of Mohr-Coulomb envelope |
| Undrained Strength | cu | UU triaxial: ½(σ₁ − σ₃) at failure |
| Angle of Failure Plane | θf | 45° + φ'/2 from horizontal |
| Pore Pressure Parameter A | A | Δu/Δσ₁ (Skempton) at failure |
| Pore Pressure Parameter B | B | Δu/Δσ₃ (≈1 for saturated soil) |
| Soil | N-value | Correlation |
|---|---|---|
| Sand (φ) | N (blows/300mm) | φ ≈ 28 + 15×(N)⁰·¹⁸ (Peck) |
| Clay (cu) | N | cu ≈ 6N kPa (approx.) |
| Relative Density | N | Dr ≈ √(N/0.85) × 100% (for sand) |
| Consistency | N | N=0-4:Very soft, 4-8:Soft, 8-15:Medium, 15-30:Stiff, 30-50:Hard, >50:Very hard |
cu is the single parameter needed. For long-term stability of clay slopes and foundations, use effective stress parameters c' and φ' from CU or CD tests. For sand, drainage is immediate — always use c'=0, φ'=φ.┌─────────────────────────────────────────────────────────────────┐
│ RANKINE & COULOMB EARTH PRESSURE THEORIES │
├─────────────────────────────────────────────────────────────────┤
│ │
│ THREE STATES OF EARTH PRESSURE: │
│ Ka: Active earth pressure (wall moves away from soil) │
│ K₀: At-rest earth pressure (no wall movement) │
│ Kp: Passive earth pressure (wall moves into soil) │
│ K₀ < Ka < 1 < Kp │
│ │
│ ═══════════════════════════════════════════════════════ │
│ RANKINE'S THEORY (smooth wall, backfill horizontal): │
│ ═══════════════════════════════════════════════════════ │
│ Active: │
│ Ka = (1 − sinφ) / (1 + sinφ) = tan²(45° − φ/2) │
│ σa = Ka × γ × z (triangular distribution) │
│ Pa = ½ × Ka × γ × H² (total force per unit width) │
│ Acting at H/3 from base │
│ │
│ Passive: │
│ Kp = (1 + sinφ) / (1 − sinφ) = tan²(45° + φ/2) │
│ σp = Kp × γ × z │
│ Pp = ½ × Kp × γ × H² │
│ Acting at H/3 from base │
│ │
│ At-Rest: │
│ K₀ = 1 − sinφ (Jaky's formula for NC soil) │
│ K₀ = (1 − sinφ) × OCR^sinφ (for OC soil) │
│ P₀ = ½ × K₀ × γ × H² │
│ │
│ WITH SURCHARGE (q): │
│ σa = Ka × γ × z + Ka × q │
│ Pa = ½ × Ka × γ × H² + Ka × q × H │
│ │
│ SUBMERGED BACKFILL (below water table at depth d): │
│ Above WT: σa = Ka × γ × z │
│ Below WT: σa = Ka × γ × d + Ka × γ'(z−d) + γw×(z−d) │
│ (γ' = submerged unit weight) │
│ │
│ WITH COHESION (c-φ soil): │
│ σa = Ka × γ × z − 2c√Ka │
│ (Negative stress at top → tension crack depth zc = 2c/(γ√Ka))│
│ │
│ ═══════════════════════════════════════════════════════ │
│ COULOMB'S THEORY (rough wall, inclined backfill): │
│ ═══════════════════════════════════════════════════════ │
│ Accounts for wall friction (δ) and backfill inclination (β) │
│ │
│ Ka = sin²(φ+α) / [sin²α × sin(α−δ) × {1+√(sin(φ+δ)sin(φ−β) │
│ / (sin(α−δ)sin(α+β))}²] │
│ │
│ Where: │
│ α = angle of wall back face from horizontal │
│ β = angle of backfill slope │
│ δ = wall friction angle = (½ to ⅔) × φ │
│ │
│ Pa acts at an angle δ to the wall normal │
│ Point of application: H/3 from base (for triangular pressure) │
└─────────────────────────────────────────────────────────────────┘| Feature | Rankine | Coulomb |
|---|---|---|
| Wall Surface | Smooth (δ = 0) | Rough (δ > 0) |
| Backfill | Horizontal | Inclined allowed |
| Failure Surface | Planar | Curved (in reality), assumed planar |
| Backfill Shape | Planar, no surcharge complexity | Any shape, broken backfill |
| Accuracy | Conservative for active | More realistic for rough walls |
| Application | Sheet piles, smooth walls | Gravity walls, cantilever walls |
| Soil | φ (°) | Ka | K₀ | Kp |
|---|---|---|---|---|
| Loose Sand | 28 | 0.361 | 0.531 | 2.77 |
| Medium Sand | 32 | 0.307 | 0.462 | 3.26 |
| Dense Sand | 38 | 0.238 | 0.384 | 4.20 |
| Soft Clay | 0 | 1.000 | 1.000 | 1.000 |
| Stiff Clay | 20 | 0.490 | 0.657 | 2.04 |
zc = 2c/(γ×√Ka). In Rankine theory, earth pressure is parallel to the backfill surface. For passive pressure, Coulomb's theory overestimates Kp for high φ values — use logarithmic spiral or Caquot-Kerisel charts instead. The point of application of total force is at H/3 for triangular pressure distribution.┌─────────────────────────────────────────────────────────────────┐
│ TERZAGHI'S BEARING CAPACITY THEORY │
├─────────────────────────────────────────────────────────────────┤
│ │
│ ULTIMATE BEARING CAPACITY (qu): │
│ qu = c × Nc + q × Nq + ½ × γ × B × Nγ │
│ │
│ Where: │
│ c = cohesion of soil │
│ q = surcharge = γ × Df (Df = depth of foundation) │
│ γ = unit weight of soil below foundation base │
│ B = width of foundation (least dimension) │
│ Nc, Nq, Nγ = bearing capacity factors (function of φ) │
│ │
│ TERZAGHI'S BEARING CAPACITY FACTORS: │
│ Nq = e^(π×tanφ) × tan²(45° + φ/2) │
│ Nc = (Nq − 1) × cot(φ) │
│ Nγ = (Nq − 1) × tan(1.4φ) [Terzaghi's empirical] │
│ │
│ GENERAL BEARING CAPACITY (IS 6403 / Meyerhof): │
│ qu = c × Nc × Sc × Dc × Ic │
│ + q × Nq × Sq × Dq × Iq │
│ + ½γBNγ × Sγ × Dγ × Iγ │
│ │
│ Shape Factors (for rectangular footing B×L): │
│ Sc = 1 + 0.2(B/L) │
│ Sq = 1 + 0.2(B/L) │
│ Sγ = 1 − 0.4(B/L) [Sγ = 1 for square] │
│ │
│ Depth Factors: │
│ Dc = 1 + 0.2(Df/B)×tan⁻¹(Df/B) [≈ 1+0.2√(Df/B)] │
│ Dq = Dc │
│ Dγ = 1 │
│ │
│ Inclination Factors (load inclined at angle α): │
│ Ic = Iq = (1 − α/90°)² │
│ Iγ = (1 − α/φ)² │
│ │
│ NET BEARING CAPACITY: │
│ qnet = qu − q = qu − γ × Df │
│ │
│ SAFE BEARING CAPACITY: │
│ qsafe = qnet / FS + γ × Df │
│ FS = Factor of Safety = 2.5-3.0 (shear) │
│ │
│ NET SAFE BEARING CAPACITY (for settlement check): │
│ qns = qnet / FS │
│ │
│ SETTLEMENT CRITERIA: │
│ Total settlement ≤ 25-50 mm (general) │
│ Differential settlement ≤ B/300 │
│ │
│ EFFECTIVE WIDTH (eccentric loading): │
│ B' = B − 2e (one-way eccentricity) │
│ Use B' in place of B for reduced bearing capacity │
│ │
│ EXAMPLE (Strip footing, φ = 30°, c = 0): │
│ Nc = 30.14, Nq = 18.40, Nγ = 22.40 │
│ For B=2m, Df=1m, γ=18 kN/m³: │
│ qu = 0 + 18×1×18.4 + ½×18×2×22.4 = 331.2 + 403.2 = 734.4 kPa│
│ qnet = 734.4 − 18 = 716.4 kPa │
│ qsafe = 716.4/3 + 18 = 257 kPa │
└─────────────────────────────────────────────────────────────────┘| Type | Depth | Use Case |
|---|---|---|
| Shallow (Df ≤ B) | Df ≤ B | Firm soil near surface, light loads |
| Spread Footing | 1-3 m | Columns, walls |
| Combined Footing | 1-3 m | Two or more columns close together |
| Raft/Mat | < 3 m | Heavy loads, weak/poor soil, covers entire area |
| Deep Foundation | Df > B | Weak soil at surface, firm layer deep |
| Pile Foundation | > 5-60 m | Transfer load to firm stratum below |
| Well Foundation | > 5 m | Bridges, waterfront structures (caissons) |
| Aspect | Detail |
|---|---|
| Plate Size | 300-750 mm (usually 300 mm or 450 mm) |
| Method | Incremental loading with settlement observation |
| Ultimate Load | Load at which settlement shoots up rapidly |
| Settlement Criterion | Failure at s = 1/5th of plate width |
| Scale Effect | For clay: Bsf/Bp; for sand: Bsf/Bp → limited accuracy |
| Safe Bearing | qu / FS (from plate test, extrapolated to footing size) |